In the following, we will assume that \((\mathcal {Y},\mathcal {D})\)  is a smooth pair and \(\mathcal {Y}\) 
is also a projective Deligne-Mumford stack with a coarse moduli scheme \(Y\)  and a polarization \((\mathcal {E}_{\mathcal {Y}},\mathcal {O}_{Y}(1))\) , and \(\pi :\mathcal {X}\rightarrow \mathbb {A}^1\)  is a simple degeneration and also a family of projective Deligne-Mumford stacks with a coarse moduli scheme \(X\)  and a relative polarization \((\mathcal {E}_{\mathcal {X}},\mathcal {O}_{X}(1))\) . Then \(\mathcal {X}_{0}[k]\)  and \(\mathcal {Y}[k]\)  are projective Deligne-Mumford stacks by the projective assumption, the projectivity of the bubble component \(\Delta \)  in [[1]}, Proposition 4.13] and the uniqueness of a coarse moduli space (as a categorical quotient) via gluing (see [[2]}, Lemma 3.2] or [[3]}, Section 2]) for any \(k\ge 0\) . Due to properties \((ii)\)  in both Proposition REF  and Proposition REF , one can actually show that two standard families \(\pi _{k}:\mathcal {X}(k)\rightarrow \mathbb {A}^{k+1}\)  and \(\pi _{k}: \mathcal {Y}(k)\rightarrow \mathbb {A}^k\)  are families of projective Deligne-Mumford stacks for any \(k\ge 0\)  as follows. Using Proposition REF  and Proposition REF  again, \(\mathcal {X}(k)\)  and \(\mathcal {Y}(k)\)  are separated Deligne-Mumford stacks of finite type, and hence have their coarse moduli spaces respectively. Furthermore, \(\mathcal {X}(k)\)  and \(\mathcal {Y}(k)\)  have projective coarse moduli schemes over \(\mathbb {A}^{k+1}\)  and \(\mathbb {A}^k\)  respectively by their groupoid construction and [[4]}, Theorem 5.3 and Proposition 4.7]. Using [[5]}, Section 5] or [[6]}, Theorem 2.16-(2)], \(\mathcal {X}(k)\)  (resp. \(\mathcal {Y}(k)\) ) possesses a generating sheaf \(p^*\mathcal {E}_{\mathcal {X}}\)  (resp. \(p^*\mathcal {E}_{\mathcal {Y}}\) ) via the natural map \(p:\mathcal {X}(k)\rightarrow \mathcal {X}\)  (resp. \(p:\mathcal {Y}(k)\rightarrow \mathcal {Y}\) ). This completes the proof of projectivity of the above two families. Next, we also assume that \(\dim \mathcal {X}_{t}=\dim \mathcal {Y}=3\)  for any \(t\in \mathbb {A}^1\) .
